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Fully gapped d-wave superconductivity in CeCu2Si2Guiming Panga,b, Michael Smidmana,b, Jinglei Zhanga,b, Lin Jiaoa,b,1, Zongfa Wenga,b, Emilian M. Nicac,d,e, Ye Chena,b,Wenbing Jianga,b, Yongjun Zhanga,b, Wu Xiea,b, Hirale S. Jeevana,b,f, Hanoh Leea,b, Philipp Gegenwartf,Frank Steglicha,b,g, Qimiao Sic, and Huiqiu Yuana,b,h,1

aCenter for Correlated Matter, Zhejiang University, Hangzhou 310058, China; bDepartment of Physics, Zhejiang University, Hangzhou 310027, China;cDepartment of Physics and Astronomy, Rice University, Houston, TX 77005; dDepartment of Physics and Astronomy, University of British Columbia,Vancouver, BC, V6T 1Z1, Canada; eQuantum Materials Institute, University of British Columbia, Vancouver, BC, V6T 1Z1, Canada; fExperimental Physics VI,Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany; gMax Planck Institute for Chemical Physics of Solids,01187 Dresden, Germany; and hCollaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

Edited by T. Maurice Rice, ETH Zurich, Zurich, Switzerland, and approved April 5, 2018 (received for review November 21, 2017)

The nature of the pairing symmetry of the first heavy fermionsuperconductor CeCu2Si2 has recently become the subject of con-troversy. While CeCu2Si2 was generally believed to be a d-wavesuperconductor, recent low-temperature specific heat measure-ments showed evidence for fully gapped superconductivity, con-trary to the nodal behavior inferred from earlier results. Here,we report London penetration depth measurements, which alsoreveal fully gapped behavior at very low temperatures. To explainthese seemingly conflicting results, we propose a fully gappedd + d band-mixing pairing state for CeCu2Si2, which yields verygood fits to both the superfluid density and specific heat, aswell as accounting for a sign change of the superconductingorder parameter, as previously concluded from inelastic neutronscattering results.

heavy fermions | CeCu2Si2 | multiband superconducting pairing |superconducting order parameter | penetration depth

The structure of the superconducting order parameter hasbeen frequently studied, due to its close relationship with the

underlying pairing mechanism. While the conventional electron–phonon pairing mechanism typically leads to s-wave states withfully opened gaps and a constant sign over the Fermi surface (1),unconventional superconductors with different pairing mecha-nisms often form states with a sign-changing order parameter(2, 3). For instance, cuprate and many Ce-based heavy fermionsuperconductors are generally believed to be d -wave super-conductors, with nodal lines in the energy gap on the Fermisurface (4–6). On the other hand, in the high-temperature iron-based superconductors, an s± state has been proposed, with achange of sign of the gap function between disconnected Fermisurface pockets, but the energy gap remains nodeless (7). Inthis context, the surprising recent discovery (8–10) of evidencefor fully gapped superconductivity in the first heavy fermionsuperconductor CeCu2Si2 (11) requires further attention.

Superconductivity in CeCu2Si2 occurs in close proximity tomagnetism. Samples with either superconducting (S type), anti-ferromagnetic (A type), or competing phases (A/S type) areobtained via slight tuning of the composition within the homo-geneity range (12). High-pressure measurements of CeCu2

(Si1−xGex )2 reveal two distinct superconducting domes, onecentered around an antiferromagnetic (AFM) instability atambient/low pressure and another near a valence instability athigh pressure (13). The close proximity of superconductivity toan AFM instability suggests that in CeCu2Si2 it is driven by thecorresponding quantum criticality. Inelastic neutron scattering(INS) measurements clearly indicate that the Cooper pairingis associated with a damped propagating paramagnon mode atthe incommensurate ordering wavevector QAF of the spin-densitywave (SDW) order nearby in the phase diagram (14). The largeintensity of the low-energy spin excitation spectrum at QAF,which reveals a spin gap in the superconducting state, as well asa pronounced peak well inside the superconducting gap 2∆≈5kBTc (14, 15), implies a sign change of the pairing function

between the two regions of the Fermi surface spanned by QAF

(16, 17). The absence of a coherence peak and the∼T 3 temper-ature dependence of the spin-lattice relaxation rate [1/T1(T )]in Cu nuclear quadrupole resonance (NQR) measured above100 mK further suggested an unconventional superconductingorder parameter with line nodes in the gap structure (15, 18).Angle resolved resistivity measurements at 40 mK indicate a 4-fold modulation of the upper critical field Hc2, as expected for ad -wave gap with dxy symmetry (19), while a sign change spanningQAF is compatible with dx2−y2 pairing symmetry (16). Therefore,CeCu2Si2 behaves as an even-parity d -wave superconductor,whose gap structure has yet to be determined.

However, a recent specific heat investigation reported expo-nential behavior of C (T )/T at very low temperatures, suggest-ing fully gapped superconductivity in CeCu2Si2 (8). Followingthis work, scenarios of multiband superconductivity with a strongPauli paramagnetic effect, loop-nodal s± superconductivity, ands++ pairing with no sign change were proposed (9, 10, 20, 21).Furthermore, scanning tunneling spectroscopy down to 20 mKalso hints at a multigap order parameter (22). Indeed electronicstructure calculations reveal that multiple bands cross the Fermilevel (8, 23), and renormalized band structure calculations showthat the dominant heavy band (with m∗/me ≈ 500) leads toFermi surface sheets mainly consisting of warped cylinders along

Significance

Identifying the gap structure of superconductors is vitalfor understanding the underlying pairing mechanism of theCooper pairs. The first heavy fermion superconductor to bediscovered, CeCu2Si2, was thought to be a d-wave supercon-ductor with gap nodes, until recent specific heat measure-ments provided evidence that the gap is fully open acrossthe Fermi surface. We propose a resolution to this puzzlefrom measurements of the London penetration depth, whichgive further evidence for fully gapped superconductivity. Weanalyze the data using a d-wave band-mixing pairing model,which leads to a fully open superconducting gap. Our modelaccounts well for the penetration depth and specific heatdata, while reconciling the nodeless and sign-changing natureof the gap function.

Author contributions: H.Y. designed research; G.P., M.S., J.Z., L.J., Z.W., E.M.N., Y.C., W.J.,Y.Z., W.X., H.S.J., H.L., P.G., and H.Y. performed research; G.P., M.S., J.Z., L.J., E.M.N., F.S.,Q.S., and H.Y. analyzed data; and G.P., M.S., L.J., E.M.N., F.S., Q.S., and H.Y. wrote thepaper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).1 To whom correspondence may be addressed. Email: hqyuan@zju.edu.cn or phyman21@gmail.com.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1720291115/-/DCSupplemental.

Published online May 8, 2018.

www.pnas.org/cgi/doi/10.1073/pnas.1720291115 PNAS | May 22, 2018 | vol. 115 | no. 21 | 5343–5347

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the c axis (23). The aforementioned discrepancies between thepairing symmetries deduced from different measurements showthat the superconducting order parameter of CeCu2Si2 is poorlyunderstood. A particular puzzle is how to reconcile the fullygapped behavior with the previous evidence for a sign-changingorder parameter and nodal superconductivity. Here we probethe superconducting gap symmetry by measuring the tempera-ture dependence of the London penetration depth and propose ascenario of a fully gapped d + d band-mixing pairing state, whichreconciles all of the seemingly contradictory results.

ResultsResistivity and Specific Heat. The samples were characterizedusing resistivity and specific heat measurements, as shown forthe S -type sample in Fig. 1A. The residual resistivity of the S -type sample in the normal state just above Tc is ρ0 ≈ 40 µΩ·cm,and a superconducting transition is observed, onsetting around0.65 K and reaching zero resistivity at about 0.6 K. The transi-tion width of ≈ 0.05 K is in line with recent reports (19). Thespecific heat also shows a superconducting transition with Tc ≈0.64 K, similar to previous results (8). The A/S -type sample(Fig. 1B) displays a superconducting transition, onsetting around0.62 K, with a lower residual resistivity of ρ0≈ 12µΩ·cm. Thespecific heat shows both an AFM transition at TN ≈ 0.7 K and asuperconducting transition at Tc ≈ 0.53 K.

Temperature Dependence of the Penetration Depth. Measurementsof the change of the London penetration depth ∆λ(T ) =λ(T )−λ(0) for the S -type sample are displayed in Fig. 2A. As shown inthe Inset, a sharp superconducting transition is clearly observed,with an onset at around 0.62 K. To probe the superconductinggap structure, we analyzed the behavior of ∆λ(T ) at low tem-peratures, and the results are shown in the main panel. The datawere fitted with the exponential temperature dependence for afully open gap, ∆λ(T ) = AT−

12 e−∆(0)/kBT + B , where ∆(0)

is the gap magnitude at zero temperature and the constant Ballows for some variation in the extrapolated zero temperaturevalue. The fitting was performed up to 0.12 K (≈Tc/5), and asshown by the solid line in Fig. 2A, the model can account forthe data with a gap of ∆(0) = 0.48kBTc . A similar gap valueof ∆(0) = 0.58kBTc is obtained from a corresponding fit forthe A/S -type sample, as displayed in the main panel of Fig.2B. The small gap values in both cases means that ∆λ(T ) onlysaturates at very low temperatures. The results indicate similarsuperconducting properties of the S - and A/S -type samples andare consistent with the fully gapped superconductivity reportedfor an S -type single crystal in ref. 8.

The penetration depth of the S - and A/S -type samplescould also be described by a power law dependence ∼Tn (SI

A B

Fig. 1. Specific heat as C(T)/T and resistivity ρ(T) of (A) S-type and (B) A/S-type CeCu2Si2.

A

B

Fig. 2. The change in London penetration depth ∆λ(T) at low tempera-ture for an (A) S-type and (B) A/S-type sample of CeCu2Si2. The solid linesshow fits to a fully gapped model described in the text, while the dashedlines show fits to a power law temperature dependence of ∆λ(T)∼ Tn. Thedata across the whole temperature range of the superconducting states aredisplayed in the Inset of A. The Inset of B shows n when the data are fittedwith ∆λ(T)∼ Tn up to a temperature Tup.

Appendix), with n = 2.24 and 2.43, respectively, when fittingfrom the base temperature to 0.12 K. For line nodes in the super-conducting gap in the presence of impurity scattering, ∆λ(T )may show quadratic behavior at low temperatures, which crossesover to linear behavior at an elevated temperature (24). To checkhow the exponent n evolves with temperature, we also fittedwith the power law expression from the base temperature upto a range of temperatures Tup , and the dependence of n onTup is shown in Fig. 2 B, Inset. It can clearly be seen that forboth samples, n increases with decreasing Tup , with n > 2. Thisindicates that the true low-temperature behavior is not a ∼T 2

dependence, as expected for a dirty nodal superconductor, but nincreases as expected for superconductivity exhibiting a full gap.Therefore, both the specific heat and ∆λ(T ) data are consistentwith fully gapped superconductivity at very low temperatures.

Analysis of the Superfluid Density. The superfluid density was cal-culated using ρs(T ) = [λ(0)/λ(T )]2 and is displayed for bothS - and A/S -type samples in Fig. 3, where λ(0) = 2000 A (25).The superfluid density was fitted following the method of ref.26, for a gap ∆k (T ) integrated over a cylindrical Fermi surface.The superfluid density data were fitted with an isotropic s-wavemodel with a gap ∆(T ) = ∆(0) tanh

[1.82

(1.018

(TcT− 1))0.51

](27) as well as a d -wave model with line nodes (∆k (T ,φ) =∆(T ) cos 2φ, φ = azimuthal angle). As shown in Fig. 3A, a

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Fig. 3. Superfluid density of CeCu2Si2 fitted with various models. Fits forthe S-type sample are shown for (A) two fully open gaps, as well as s-, andd-wave models denoted by solid, dotted, and dashed lines, respectively, and(B) a d + d band-mixing pairing model. Fits for the A/S-type sample areshown for (C) two fully open gaps, as well as s-, and d-wave models denotedby solid, dotted, and dashed lines, respectively, and (D) a d + d band-mixingpairing model.

single-band isotropic s-wave gap cannot account for the data ofthe S -type sample, in contrast to the low-temperature ∆λ(T )data discussed above (Fig. 2A). The single-band d -wave modelshows reasonable agreement above 0.5 Tc , but the agreementis poor at low temperatures, since for this model ρs(T ) is lin-ear but the data are not. The agreement with the d -wave modelat higher temperatures is consistent with the previously reportedevidence for d -wave superconductivity. The data were also fit-ted using a two-gap s-wave model (27), and this gives reasonableagreement. The fitted gap values are ∆1(0) = 1.96kBTc and∆2(0) = 0.75kBTc , with a fraction for the larger gap of x1 = 0.74.Both gap values are slightly larger than the ones obtained forthe two-gap model in ref. 8. Similarly, neither the s- nor d -wavesingle-band models could describe the superfluid density of theA/S -type sample at low temperatures (Fig. 3C), but the datacould be fitted using a two-gap model with ∆1(0) = 2.0kBTc ,∆2(0) = 0.75kBTc , and x1 = 0.74.

On the other hand, it is difficult to reconcile an s-wave modelwith the evidence for a sign-changing gap function, as con-cluded from the INS response, where a sharp spin resonanceforms at the edge of a spin gap well inside the superconductinggap (14). Moreover, the incommensurate ordering wave vec-tor of the nearby SDW (QAF) is identical to the nesting wavevector spanning the flat parallel parts of the warped cylinders(28). This shows that there is a sign change of the pair wavefunction inside the dominating heavy-fermion band, which isincompatible with a nodeless s± pairing state (21). On the otherhand, if there is no sign change of the gap function across theFermi surface, ∆k∆k+q = |∆k||∆k+q |, the coherence factor inthe spin susceptibility χ′′(q,ω) is vanishingly small (29, 30). Con-sequently, in this case, the spin spectrum will not have a sharppeak, although there may be a broad enhancement of the spec-tral weight above 2∆ (31). However, when there is a change ofsign of the gap function between regions of the Fermi surfaceconnected by q = QAF, there is an enhanced coherence factorsince ∆k∆k+q =−|∆k||∆k+q |. This gives rise to a sharp peak inχ′′(q,ω) below 2∆, leading to the conclusion that there must be asign-changing order parameter in CeCu2Si2 (16, 17). By a similarargument, the lack of a coherence peak in NQR measurements

also strongly disfavors superconductivity without a sign reversal(15, 18, 32). Furthermore, given the strong Coulomb repulsionin Ce-based heavy fermion superconductors, the order param-eter must be anisotropic with a sign change, without runninginto the issue of a large µ∗ (33). In other words, the f -electronshave a Coulomb repulsion that is much larger than their effec-tive Fermi energy, and they must avoid each other, therebyexcluding any sign-preserving pairing function. In such stronglycorrelated superconductors, even anisotropic and sign-changingpairing states can be robust against disorder (33). Indeed, poten-tial scattering [due to a site exchange between Cu and Si of lessthan 1% within the homogeneity range (34)], which enhances theresidual resistivity by a factor of ≈4, apparently has an almostnegligible influence on Tc (cf., the resistivity results on the S andA/S samples in Fig. 1). Also, recent experiments on electron-irradiated samples revealed only a minor change of Tc (9). Atthe same time, just like in the cuprates, the effect of substitu-tional disorder on Tc is known to be site- and size-dependent(35). For CeCu2Si2, the superconducting Tc was found to beextremely sensitive to nonmagnetic substitutions on the Cu site:For example, Rh, Pd, and Mn substitution for Cu at a level of≈1% fully suppresses superconductivity (35), which is impossibleto account for in the scenario of an s-wave state without a signchange of the order parameter. Further studies are needed todevelop a detailed understanding of all these observations.

In the present work, we consider a pairing function that, byanalogy with an sτ3 pairing state (36), has an effective gap asa result of intraband pairing with dx2−y2 symmetry and inter-band pairing with dxy symmetry; our pairing function preservesboth the fully gapped nature and order parameter sign changealong QAF on a single nested Fermi surface. The sτ3 pairing statewas introduced in the context of the iron-based superconductors(SI Appendix) (36), as part of the studies about orbital-selectivesuperconducting pairing (37–40). There, the pairing function hasthe form ∆∼ sx2y2(k)× τ3 (“sτ3”), as a product of an s-waveform factor and a Pauli matrix in the dxz , dyz orbital subspace.For that case, the interorbital mixing in the dispersion part ofthe Hamiltonian ensures that in the band basis the pairing isequivalent to a superposition of intra- and interband compo-nents with dx2−y2(k) and dxy(k) form factors, respectively. Theresulting quasiparticle spectrum acquires a nonvanishing |∆(k)|2contribution, as the two components are added in quadrature,ensuring a full gap on the whole Fermi surface with a signchange of the intraband component of the gap function. It isalso shown how this pairing channel can be stabilized within aself-consistent five-orbital model, with a full gap and a resonancein the spin-excitation spectrum. Similar to the case consideredin ref. 36, QAF of CeCu2Si2 will connect two parts of the Fermisurface with a sign change in the intraband component of thegap function (see Fig. 4), thereby generating an enhanced spinspectral weight just above a threshold energy E0 = 3.9kBTc (14),inside the superconducting gap 2∆1≈ 5 kBTc (see below andref. 15).

In the following, we apply a simplified model for thegap structure to CeCu2Si2, given by summing contributionsfrom the two d -wave states in quadrature, with ∆(T ,φ) =

[(∆1(0) cos(2φ))2 + (∆2(0) sin(2φ))2]12 δ(T ), where δ(T ) is the

gap temperature dependence from Bardeen–Cooper–Schrieffertheory (1), which we used previously. In general, a d + d band-mixing pairing can introduce corrections to the gap given above,due to the nondegeneracy of the bands throughout the Bril-louin zone, which would then lead to an extra parameter, theband splitting. In the following, we will show that the data canbe well fit by the simple function without this extra parameter.Although the dxy and dx2−y2 states each have two line nodes,the nodes of the two states are offset by π/4 in the kx − ky plane,and as a result, the gap function is nodeless everywhere on a

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Fig. 4. An illustration of the warped parts of the cylindrical Fermi surfaces(red) in CeCu2Si2 at particular values of kz, corresponding to the nestingportions of the 3D Fermi surface, as well as additional smaller pockets (blue)projected onto the kx − ky wavevector plane (23). The component Q‖

AF ofthe antiferromagnetic wavevector QAF projected into the same wavevectorplane connects the parts of the heavy Fermi surface with a sign change in theintraband pairing component. The corresponding Fermi surface and nestingwavevector τ = QAF in the 3D space are those displayed in figure 3b ofref. 28.

cylindrical Fermi surface. It should also be noted that for thismodel the same ρs(T ) is calculated upon exchanging ∆1(0) and∆2(0). The superfluid density of the S -type sample was fittedusing this model, and the results are shown in Fig. 3B. It can beseen that such a model can also fit the data well, with gap param-eters of ∆1(0) = 2.5kBTc and ∆2(0) = 0.58kBTc . In Fig. 3D,ρs(T ) for the A/S -type sample is equally well fitted, with sim-ilar parameters of ∆1(0) = 2.54kBTc and ∆2(0) = 0.6kBTc . Thevalues of the larger gap agree almost perfectly with the gap valueobtained from Cu-NQR measurements at higher temperatures(15). Furthermore, this model only uses two fitting parameters,while the two-band s-wave model of ref. 8 needs three.

Analysis of the Temperature Dependence of the Specific Heat. Wealso reanalyzed the specific heat data digitized from ref. 8 usingthe d + d band-mixing pairing model. As shown in Fig. 5, thedata can also be well described using this model, with fittedparameters ∆1(0) = 2.08 kBTc and ∆2(0) = 0.55 kBTc . Thevalue of the small gap is similar to that obtained from the super-fluid density fit, while the large gap is smaller in comparison. Itshould be noted that the calculated superfluid density requires anestimation of G/λ(0), where G is a calibration constant for thetunnel diode oscillator (TDO) method, and the small differencesin the gap values from the fits may arise due to uncertainties inthis value.

Discussion and SummaryBoth the superfluid density and specific heat results are highlyconsistent with a model of the d + d band-mixing pairing state,which most importantly also explains the sign change of thesuperconducting order parameter. Although the fully gappednature of the pairing state means that the density of states N (E)is zero at low energies, N (E) is nearly linear above the small gap,much like for pairing states with line nodes. This is also consis-tent with the literature results showing d -wave superconductivity(15, 18), which were not obtained at low enough temperaturesto observe clear evidence for fully gapped behavior. The lack of

a coherence peak below Tc in the Cu-NQR 1/T1(T ) measure-ments (15, 18, 32) can also not be accounted for by a two-gaps-wave model but is readily taken into account by the anisotropicd + d state, which changes sign along QAF across the Fermi sur-face. We note that the effective gap corresponding to a d + dband-mixing pairing is formally identical to one obtained froma d + id pairing and the good fits to ρs(T ) and the specificheat are also consistent with this pairing. In the d + id state,time-reversal symmetry would be broken and although no clearexperimental indication of a time-reversal symmetry-breakingsuperconducting state was found from µSR measurements ofA/S -type samples (41), this requires further study. By contrast, ad + d band-mixing pairing is invariant under time reversal, whilegenerating the expected sign change.

From a theoretical perspective, unconventional superconduc-tors in the presence of strong correlations are generally expectedto be robust against disorder (33). This has been demonstrated inmodels for strongly correlated superconductivity driven by short-range spin-exchange interactions (42, 43). The sτ3 pairing state(36) arises in a similar fashion and is also expected to be robustagainst disorder. Because the 4f electrons in heavy fermionsystems undoubtedly have strong correlations, the d + d band-mixing pairing state proposed for CeCu2Si2 should be similarlyrobust to disorder, except for atomic substitutions (35, 44).

In conclusion, we have studied the change of penetrationdepth ∆λ(T ) and normalized superfluid density ρs(T ) of theheavy fermion superconductor CeCu2Si2 (both A/S - and S -type samples). The behavior of ∆λ(T ) at very low temperaturesagrees with fully gapped superconductivity, as concluded fromspecific heat measurements (8). We demonstrate that a nodelessd + d band-mixing pairing state can account for the tempera-ture dependence of both the superfluid density and specific heat.This state has the necessary sign change of the superconductingorder parameter along QAF on the heavy Fermi surface deducedfrom INS (14) and is consistent with the lack of a coherence peakin 1/T1(T ). The model also explains the consistency of d -wavesuperconductivity at higher temperatures, previously reportedfrom 1/T1(T ) measurements (15, 18, 32). We therefore proposethis d + d band-mixing pairing state to be the superconductingorder parameter of CeCu2Si2. Given that this is also a strong can-didate pairing state for FeSe-based superconductors (36), such apairing model may well be applicable to a wider range of fullygapped unconventional superconductors, including the case of asingle CuO2 layer (45).

Fig. 5. Specific heat of S-type CeCu2Si2 digitized from ref. 8. The solid lineshows a fit to the d + d band-mixing pairing model.

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Materials and MethodsCeCu2Si2 single crystals were synthesized by a modified Bridgman tech-nique using a self-flux method (46). The temperature dependence of theLondon penetration depth shift ∆λ(T) = G∆f(T) was measured down toabout 40 mK using a TDO-based technique (47), where ∆f(T) = f(T)− f(0),f(T) is the resonant frequency of the TDO coil, and G is a calibrationconstant (48).

ACKNOWLEDGMENTS. We thank R. Yu, S. Kirchner, G. Zwicknagl,P. Coleman, D. J. Scalapino, O. Stockert, S. Kittaka, and M. B. Salamon forvaluable discussions. The work at Zhejiang University has been supported

by National Key R&D Program of China Grants 2017YFA0303100 and2016YFA0300202; National Natural Science Foundation of China GrantsU1632275, 1147425, and 11604291; and the Science Challenge Project ofChina Project TZ2016004. The work at Rice University has been supportedby NSF Grant DMR-1611392; Robert A. Welch Foundation Grant C-1411; aQuantEmX grant from the Institute for Complex Adaptive Matter (ICAM)and Gordon and Betty Moore Foundation Grant GBMF5305 (to Q.S.); andthe Center for Integrated Nanotechnologies, a US DOE Basic Energy Sciences(BES) user facility. The work is also supported by the Sino-German Coopera-tion Group on Emergent Correlated Materials (GZ1123). Q.S. acknowledgesthe hospitality of the Aspen Center for Physics (NSF Grant PHY-1607611) andthe University of California, Berkeley.

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